Gain of OP-amp circuit - Inverting configuration
We wish to find the DC gain \(A_v = \frac{V_o}{V_{in}}\) of this inverting amplifier circuit.
To do this, write the node equation for \(V_A\). $$\frac{V_A-V_{in}}{R_1} + \frac{V_A-V_o}{R_f} = 0 \tag{1}$$ We assume an ideal OP-amp. Since there is negative feedback we can use the principle of a virtual short. The virtual short causes \(V_+ = V_-\). Since \(V_+\) is grounded, we have that \(V_+ = V_- = V_A = 0\text{V}\). The node equation in (1) now becomes $$\frac{-V_{in}}{R_1}-\frac{V_o}{R_f} = 0 \Leftrightarrow \tag{2}$$ $$\frac{V_o}{R_f} = -\frac{V_{in}}{R_1} \Leftrightarrow \tag{3} $$ $$\boxed{A_v = \frac{V_o}{V_{in}} = -\frac{R_f}{R_1}} \tag{4}$$ And we have found the DC gain of this inverting OP amp configuration. If we wanted a gain of -2 a possible resistor combination could be \(R_f = 2\text{k}\Omega, \: R_1 = 1\text{k}\Omega \) because \(A_v = -\frac{R_f}{R_1} = -\frac{2\text{k}\Omega}{1\text{k}\Omega} = -2 \).
This method of writing up node equations, and then using the virtual short principle can be used for every linear OP-amp circuit. It is very efficient.
To do this, write the node equation for \(V_A\). $$\frac{V_A-V_{in}}{R_1} + \frac{V_A-V_o}{R_f} = 0 \tag{1}$$ We assume an ideal OP-amp. Since there is negative feedback we can use the principle of a virtual short. The virtual short causes \(V_+ = V_-\). Since \(V_+\) is grounded, we have that \(V_+ = V_- = V_A = 0\text{V}\). The node equation in (1) now becomes $$\frac{-V_{in}}{R_1}-\frac{V_o}{R_f} = 0 \Leftrightarrow \tag{2}$$ $$\frac{V_o}{R_f} = -\frac{V_{in}}{R_1} \Leftrightarrow \tag{3} $$ $$\boxed{A_v = \frac{V_o}{V_{in}} = -\frac{R_f}{R_1}} \tag{4}$$ And we have found the DC gain of this inverting OP amp configuration. If we wanted a gain of -2 a possible resistor combination could be \(R_f = 2\text{k}\Omega, \: R_1 = 1\text{k}\Omega \) because \(A_v = -\frac{R_f}{R_1} = -\frac{2\text{k}\Omega}{1\text{k}\Omega} = -2 \).
This method of writing up node equations, and then using the virtual short principle can be used for every linear OP-amp circuit. It is very efficient.